In the case of OFDM systems which employ wideband pilot signals, an initial coarse estimate of the CIR is obtained using a regularized version of the MLS algorithm. Here the subscript j denotes the jth receive antenna, n denotes the nth OFDM symbol. For example, consider a system with single transmit antenna and multiple receive antenna. After FFT of the received OFDM symbol at the receiver, bi,n, the received data at pilot locations at the nth OFDM symbol is given by
                              b                      j            ,            n                          =                                            A                              n                ,                L                                      ⁢                          h                              j                ,                n                                              +                      N                          j              ,              n                                +                                    ∑                              i                =                1                            M                        ⁢                                          A                                  n                  ,                  L                  ,                  i                                            ⁢                              h                                  j                  ,                  n                  ,                  i                                                                                        (        1        )            where hj,n is the desired CIR at the jth receive antenna in the nth OFDM symbol and hj,n,i; is the CIR of the ith interferer. Here bj,n is the received samples (after FFT) at the pilot locations, An,L=SnFL and An,L,i=Sn,iFL with Sn being a P×P diagonal matrix where the diagonal entries correspond to pilots and/or data for the user and FL is the DFT matrix of dimension P×L where the columns of the DFT matrix correspond to multipath locations of hj,n and the rows correspond to pilot positions. Similarly Sn,j is a P×P diagonal matrix where the diagonal entries correspond to pilots and/or data of the interfering user. Nj,n is a P×1 vector corresponding to thermal noise at the pilot locations. The CIR comprises of L multipath components and has the form
                                          h            j                    ⁡                      (            k            )                          =                              ∑                          i              =              1                        L                    ⁢                                    α              i                        ⁢                          δ              ⁡                              (                                  k                  -                  i                                )                                                                        (        2        )            where αi is a zero mean complex Gaussian variable and δ(.) is the Dirac delta operator. The path delays are assumed to sample spaced however the proposed method can be used even in the case of a non sample spaced model. In the case of a non-sample spaced model, the equivalent sample spaced version is looked at wherein due to leakage the number of non-zero sample spaced taps will be higher than the actual number of non-sample spaced channel taps and a DFT matrix of appropriate dimension is used. Generally L will be much smaller than the cyclic prefix (CP) of length LCP.
One of the simplest channel estimation schemes for OFDM system is the modified least squares (MLS) scheme which exploits the fact that the CIR length L is at most equal to the CP length LCP. This leads to the following channel estimate given byĥj,n=(An,LCPHAn,LCP)−1An,LCPHbj,n  (3)where An,LCP=SnFLCP with FLCP corresponding to the DFT matrix of dimension P×LCP. In low SINR scenarios, the MLS estimator has a very poor performance due to the effect of noise and interference. The MSE of MLS estimator is approximately equal to σ2LCP/P where σ2 is the variance of sum of interference and noise term at a subcarrier and P is the number of pilots. The MSE value is very high at low SINR, i.e. for example in LTE system with LCP=80 and P=100 the corresponding MSE value is σ2LCP/P≈σ2. Furthermore, in case of MLS, the matrix inverse (AHn,LCPAn,LCP)−1 is ill-conditioned due to the presence of guard tones/virtual carriers in the OFDM symbols and hence will lead to a highly smeared CIR estimate in the time-domain.
One way to combat this is to use regularization theory and estimate the CIR as a regularized LS problem. The regularized MLS solution is given byĥj,n=(An,LCPHAn,LCP+γILCP)−1An,LCPHbj,n  (4)While there has been a lot of papers/patents which use diagonal loading/regularization to reduce the ill-conditioned nature of (AHn,LCPAn,LCP)−1 most of them suggest a ad-hoc diagonal loading values such as γ=10−5 or γ<<1/P. If the regularization term γ is not chosen properly the time domain MLS estimate still suffers from smearing and it is not possible to apply model order identification and tap detection algorithm to the time domain channel estimate.
One of the most popular model order identification methods is based on AIC/GAIC. While GAIC and AIC based methods are highly efficient they are quite computationally complex and may be difficult to implement in practice.
Let {Mk: k=1, 2, . . . , K} be a set of competing models indexed by k=1, 2, . . . ,K. Then the criterionAIC(k)=−2 log L({circumflex over (θ)}k)+2k  (7)which is minimized to choose the model Mk over the set of models is called the AIC criterion. L({circumflex over (θ)}k) is the likelihood function assuming model order k. Here the term 2k penalizes overfitting while −2 log L({circumflex over (θ)}k) penalizes underfitting. It is well known that as the number of observations tends to infinity the probability of overfitting tends to a constant greater than zero while the probability of underfitting tends to zero for the AIC. The non-zero overfitting probability is clue to fact that the term 2k which penalizes overfitting is small.
The following generalized information criterion (GAIC) given byGAIC(k)=−2 log L({circumflex over (θ)}k)+νk  (8)may outperform AIC if ν>2. However, there is no clear guideline for choosing ν and no expression given for the best ν. Based on empirical studies it is believed that values in the interval νε[2, 6] gives the best performance.
The Bayesian information criterion selects the order that minimizesBIC(k)=−2 log L({circumflex over (θ)}k)+k ln N  (9)
While the GAIC criterion and other information criterion such as Bayesian Information Criterion (BIC) are very accurate in finding the model order and channel tap position at moderate to high SINR they involve a high computational load. For example, in case of a CP of length LCP to find the model order (7) will have to be computed LCP times. Then to find the sparsity information, the last tap (estimated model order) is removed and then again finds the new the model order. If the estimated model order (last tap position) was l then again one has to compute (7) l times to find the new model order and hence find the position of the second last tap. This procedure continues till one obtains all the significant taps. The complexity of such a process depends on the number of taps and the tap locations and the number of estimated taps. Since multiple passes of the GAIC algorithm have to be done, it is time consuming and complex.
One simple solution in case of very low SINR regime, i.e., cell edge scenario is to simply assume that model order is small (since most model order methods fail to find the actual model order and the MLS solution is also extremely poor, as it implicitly assumes a highly overparametrized model by assuming that channel length is LCP) and hard fix the model order to number say gLCP with g<1 based on the rough knowledge the receiver has of the channel selectivity. It can be shown through simulations that such an approach works very well for SINRs below 2 dB since in such cases even GAIC/AIC are not able to find the correct taps locations as the high interference and noise swamp out the channel taps. In fact in most cases GAIC ends up with a model order of 1 or zero, i.e. it barely manages to locate the LOS tap and assumes that there are no other significant taps. Therefore instead of even trying to apply model order techniques one could simply assume a fixed number (sufficiently small) of taps. It can be shown that assuming that number of non-zero taps to be eight when the CP length LCP is 80 works very well for all the channel models defined for LTE, i.e. PEDA, EPEDA, PEDB, EVA and ETU tabulated in Table 1.
This method is called the fixed model order method and let the fixed value for model order be M. Now the complexity of the fixed model order is even lesser than the computational complexity of the baseline MLS algorithm since M<<LCP.
While the fixed model order method work very well compared to the baseline MLS method and the GAIC method as long as SINR is low, for example, SINR<2 dB, its performance will gradually degrade as the SINR increases in comparison to other model order method since at higher SINRs a good model order method can find out the actual taps locations and hard fixing the model order would not be a good idea.